In an earlier paper [5] of the author bisimple weakly inverse semigroups with partial identities were studied. The aim of this paper is to extend the results to a wider class of semigroups, viz: bisimple weakly inverse semigroups with partial right unitoids. It is found that an ℛ-class of weakly inverse semigroup is a right skew groupoid R = (R, P), where P is a right skew semigroup [5], P⊆R, and R is a partial semigroup satisfying certain conditions. When S is a bisimple weakly inverse semigroup with E the set of partial right unitoids, it can be shown that the ℛ-class R = (R, P) containing E, which is a right skew groupoid, satisfies the following:

(i) for any a, b ∈ R, there exists c ∈ R such that Pa ∩ Pb = Pc;

(ii) for any a ∈ R, there exists a left identity e of R such that (Pa ∩ P)e = Pa ∩ P.

This note provides yet another example of the difficulties that arise when one wants to extend the spectral theory of subnormal operators to subnormal tuples. Several basic properties of a subnormal operator Y remain true for tuples; e.g. the existence and uniqueness of its minimal normal extension N, the spectral inclusion σ(N)⊂ σ(Y)-proved for n-tuples in [4] and generalized to infinite tuples in [5]. However, neither the invariant subspace theorem nor the spectral mapping theorem in the “strong form” as in [3] is known so far for subnormal tuples.

The combinatorial investigation of graphs embedded on surfaces leads one to consider a pair of permutations (σ, α) that generate a transitive group [7]. The permutation α is a fixed-point-free involution and the pair is called a map. When this condition on α is dropped the combinatorial object that arises is called a hypermap. Both maps and hypermaps have a topological description: for maps a classical reference is [13] and for hypermaps such a description can be found in [4] and [6]; a brief account of it will be given below. However, the relationship between maps and hypermaps is not simply that the latter generalize the former. Actually, with every hypermap there is associated a map, its bipartite map, and conversely every bipartite map arises in this way. We do not enter into the details of this question; we refer the reader to the work of Walsh [16]. In this sense hypermaps are, at the same time, a generalization and a special case of maps.

A compact bordered Klein surface of genus g ≥ 2 has maximal symmetry [4] if its automorphism group is of order 12(g − 1), the largest possible. An M*-group [8] acts on a bordered surface with maximal symmetry. The first important result about these groups was that they must have a certain partial presentation [8, p. 5]. However, research has tended to focus more on the surfaces with maximal symmetry than on the M*-groups, and results about these groups typically deal with existence.

Let Tn denote the full transformation semigroup on the finite set = {1, 2, … n}, that is the set of all mappings from to , with function composition as the semigroup operation. In this paper algorithms are introduced to solve equations such as axmb = c and ax = xb (a, b, c ∊ Tn), which employ a representation of members of Tn as special directed graphs.

Let ℝn be the n-dimensional Euclidean space with the usual norm denoted by |·| In what follows 蒆 will denote an open bounded subset of ℝn, and its closure.

For α ∊(0,1], is the space of all functions such that:

is called the Holder space with exponent a and is a Banach space when endowed with the norm:

where ‖u‖∞ is, as usual, defined by:

Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R-module is finitely embedded.

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.

Among such forms, let . The Epstein zeta function of f is denned to be

Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,

We prove a corresponding result for theta functions. For real α > 0, let

This function satisfies the functional equation

(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

If X is a Klein surface (KS) with boundary, of algebraic genus p, and Φ is an automorphism of order N, May [8] proved that N ≤ 2p + 2 when X is orientable and p is even, and N ≤ 2p otherwise.

He proved also that the unique topological type of an orientable KS having an orientation-preserving automorphism of maximum order is a surface with one boundary component when p is even, with two boundary components when p is odd.

The study of special radicals was begun by Andrunakievič [1]. A class of prime rings is called special if it is hereditary and closed under prime extensions. The upper radicals determined by special classes are called special. In later works Andrunakievič and Rjabuhin [2] and [3] defined the concept of a special class of modules.

A universal algebra A is said to have the basis property (BP) if any two minimal generating sets (bases) for a subalgebra of A have the same cardinality. This property was studied by the author for inverse semigroups in [5, 6]. For instance free inverse semigroups have BP. When treated as universal algebras, a classical theorem of linear algebra states that vector spaces have BP. In this paper we study BP for semigroups.

The purpose of this note is to generalize to fields of characteristic two the results obtained in [4]. We obtain necessary and sufficient conditions involving quadratic forms for certain tensor products of quaternion algebras to be division algebras.

Small [11] gave the first example of a right hereditary PI ring which is not left hereditary. Robson and Small [9] proved that a prime PI right hereditary ring is a classical order over a Dedekind domain, and hence is Noetherian (and therefore left hereditary). The authors have shown [4] that a right hereditary semiprime PI ring which is finitely generated over its center is left hereditary. In this paper we consider right hereditary PI rings T which are affine (i.e. finitely generated as an algebra over a central subfield k).